Three open envelopes with different groups of generic coin tokens. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Translate the event before choosing a formula
Let X be the number of heads in n independent flips with constant heads probability p. “Exactly k” means X = k. “At least k” means X ≥ k and includes k itself. “At most k” means X ≤ k. These phrases change which outcomes belong in the event; they are not interchangeable descriptions of the same probability.
Use a small distribution you can check
For three fair flips, the head-count probabilities are 1/8, 3/8, 3/8 and 1/8 for counts zero through three. Exactly two heads therefore has probability 3/8 = 37.5%. At least two heads includes counts two and three, giving 4/8 = 50%. At most two heads includes counts zero, one and two, giving 7/8 = 87.5%.
| Event | Included head counts | Probability |
|---|---|---|
| Exactly 2 | 2 | 37.5% |
| At least 2 | 2, 3 | 50% |
| At most 2 | 0, 1, 2 | 87.5% |
Check complements at the correct boundary
The complement of “at least k” is “at most k−1,” not “at most k.” The complement of “more than k” is “at most k.” Those one-count differences matter most in short experiments, where a single probability mass can be substantial. For p = 0 or p = 1, all probability sits at one count; the calculator handles these boundary models explicitly.
Keep count events separate from ordering
Use the Coin Flip Probability Calculator after writing the count event in symbols. Its binomial model requires independent trials and a fixed p. Read head counts versus fixed sequences if the problem names a particular order. For a run of adjacent heads anywhere in the record, move to the streak calculator; a binomial count does not require the heads to be consecutive.
Check the model behind your result
Use the Coin Flip Probability Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26